Chapter 16 Oscillatory Motion and Waves
16.6 Uniform Circular Motion and Simple Harmonic Motion
Summary
- Compare simple harmonic motion with uniform circular motion.
There is an easy way to produce simple harmonic motion by using uniform circular motion. Figure 2 shows one way of using this method. A ball is attached to a uniformly rotating vertical turntable, and its shadow is projected on the floor as shown. The shadow undergoes simple harmonic motion. Hooke’s law usually describes uniform circular motions ( [latex]{\omega}[/latex] constant) rather than systems that have large visible displacements. So observing the projection of uniform circular motion, as in Figure 2, is often easier than observing a precise large-scale simple harmonic oscillator. If studied in sufficient depth, simple harmonic motion produced in this manner can give considerable insight into many aspects of oscillations and waves and is very useful mathematically. In our brief treatment, we shall indicate some of the major features of this relationship and how they might be useful.
Figure 3 shows the basic relationship between uniform circular motion and simple harmonic motion. The point P travels around the circle at constant angular velocity [latex]{\omega}.[/latex] The point P is analogous to an object on the merry-go-round. The projection of the position of P onto a fixed axis undergoes simple harmonic motion and is analogous to the shadow of the object. At the time shown in the figure, the projection has position [latex]{x}[/latex] and moves to the left with velocity [latex]{v}.[/latex] The velocity of the point P around the circle equals [latex]{\bar{v}_{\text{max}}}.[/latex] The projection of [latex]{\bar{v}_{\text{max}}}[/latex] on the [latex]{x}[/latex] -axis is the velocity [latex]{v}[/latex] of the simple harmonic motion along the [latex]{x}[/latex] -axis.
To see that the projection undergoes simple harmonic motion, note that its position [latex]{x}[/latex] is given by
where [latex]{\theta=\omega{t}},\:{\omega}[/latex] is the constant angular velocity, and [latex]{X}[/latex] is the radius of the circular path. Thus,
The angular velocity [latex]{\omega}[/latex] is in radians per unit time; in this case [latex]{2\pi}[/latex] radians is the time for one revolution [latex]{T}.[/latex] That is, [latex]{\omega=2\pi/T}.[/latex] Substituting this expression for [latex]{\omega},[/latex] we see that the position [latex]{x}[/latex] is given by:
This expression is the same one we had for the position of a simple harmonic oscillator in Chapter 16.3 Simple Harmonic Motion: A Special Periodic Motion. If we make a graph of position versus time as in Figure 4, we see again the wavelike character (typical of simple harmonic motion) of the projection of uniform circular motion onto the [latex]{x}[/latex] -axis.
Now let us use Figure 3 to do some further analysis of uniform circular motion as it relates to simple harmonic motion. The triangle formed by the velocities in the figure a